92 V NORMED SPACES

of real numbers or over the field of complex numbers (such a space being
respectively called real and complex vector space); when the field of scalars
is not specified, it is understood that the definitions and results are valid
in both cases.* When several vector spaces intervene in the same statement,
it is understood (unless the contrary is specified) that they have the same field
of scalars. A complex vector space E can also be considered as a real vector
space by restricting the scalars to R; when it is necessary to make the distinc-
tion, we say that this real vector space E0 is underlying the complex vector
space E; if E has finite dimension n over C, E0 has dimension 2n over R.

A norm in a vector space E is a mapping (usually written x -» \\x\\, with
eventual indices to the ||. ,||) of E into the set R of real numbers, having
the following properties:

(I) ||*1| > 0 for every x e E.
(II) The relation ||*|| = 0 is equivalent to. x = 0.
(HI) ||Ax|| = |A| • ||jc|| for any * e E and any scalar A.

(IV) |l*+.y|| < ||*ll + \\y\\ for any pair of elements of E (^triangle
inequality'9).

(5.1.1) If x-* \\x\\ is a norm on the vector space E, then d(x,y) = ||* — y\\
is a distance on E such that d(x + z, y -f z) = d(x, y) andd(lx, Ay) = |A| d(x, y)
for any scalar A.

The verification of the axioms of Section 3.1 is trivial.

A normed space is a vector space E with a given norm on E; such a
space is always considered as a metric space for the distance ||* — j;||.
A Banach space is a normed space which is complete.

If E is a complex normed vector space, x-* \\x\\ is also a norm on the
underlying real vector space E0, and the metric spaces E and E0 are identical;
hence if E is a Banach space, so is E0.

Examples of Norms

(5.1.2) The examples given in (3.2.1), (3.2.2), (3.2.3), and (3.2.4) are real
vector spaces, and the distances introduced in those examples are deduced
from norms by the process of (5.1.1). The normed spaces thus defined in
examples (3.2.1) to (3.2.3) are complete by (3.20.16) and (3.14.3), hence

* The product of a scalar A and a vector x is indifferently written Xx or xX; 0 is the
neutral element of the additive group of the vector space.r point of A; given e > 0, let r > 0on, since